Chebyshev filter

Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the operating frequency range of the filter, but they achieve this with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. Type I Chebyshev filters are usually referred to as "Chebyshev filters", while type II filters are usually called "inverse Chebyshev filters". Because of the passband ripple inherent in Chebyshev filters, filters with a smoother response in the passband but a more irregular response in the stopband are preferred for certain applications. Type I Chebyshev filters are the most common types of Chebyshev filters. The gain (or amplitude) response, , as a function of angular frequency of the th-order low-pass filter is equal to the absolute value of the transfer function evaluated at : where is the ripple factor, is the cutoff frequency and is a Chebyshev polynomial of the th order. The passband exhibits equiripple behavior, with the ripple determined by the ripple factor . In the passband, the Chebyshev polynomial alternates between -1 and 1 so the filter gain alternate between maxima at and minima at . The ripple factor ε is thus related to the passband ripple δ in decibels by: At the cutoff frequency the gain again has the value but continues to drop into the stopband as the frequency increases. This behavior is shown in the diagram on the right. The common practice of defining the cutoff frequency at −3 dB is usually not applied to Chebyshev filters; instead the cutoff is taken as the point at which the gain falls to the value of the ripple for the final time. The 3 dB frequency is related to by: The order of a Chebyshev filter is equal to the number of reactive components (for example, inductors) needed to realize the filter using analog electronics.